A triple category is an n-fold category for $n=3$. That is, it is an internal category in double categories. This concept can be considered at various levels of strictness. However, there is not yet a definition in the literature of a fully weak (i.e. pseudo) triple category.

Just as any category gives rise to a double category of commutative squares, any double category gives rise to a triple category of commutative “cubes” by taking the transversal morphisms to be the same as the vertical morphisms.

Last revised on July 28, 2023 at 14:25:06.
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